A point \(y\in \mathbb {R}^n\)  is said to be inner \(\alpha \) -regular for \(A\)  if \(\varepsilon _y=(\varepsilon _y)^A=:\varepsilon _y^A\) , \(\varepsilon _y\)  being the unit Dirac measure at \(y\) ;\(\varepsilon _y^A\)  is said to be the (fractional) inner \(\alpha \) -harmonic measure of \(A\subset \mathbb {R}^n\)  at \(y\in \mathbb {R}^n\) . Being a natural generalization of the classical concept of (2-)harmonic measure [1]}, [2]}, [3]}, [4]}, \(\varepsilon _y^A\)  serves as the main tool in solving the generalized Dirichlet problem for \(\alpha \) -harmonic functions. Besides, due to the integral representation formula
\(\mu ^A=\int \varepsilon _y^A\,d\mu (y)\)  [5]},
the inner \(\alpha \) -harmonic measure \(\varepsilon _y^A\)  is a powerful tool in the investigation of the inner balayage \(\mu ^A\)  for arbitrary \(\mu \)  (see [5]}).
the set of all those \(y\)  is denoted by \(A^r\) . Then \(A^r\subset \overline{A}\) , since obviously \(\varepsilon _x^A\in \mathcal {E}^+_\alpha \)  for all \(x\notin \overline{A}\) .
The other points of \(\overline{A}\) , i.e.
\(y\in \overline{A}\setminus A^r=:A_I,\) 
